Phase space Hamiltonian mapping

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

Gray, Rice, and Davis [12] developed an alternative RRKM (ARRKM) theory in an attempt to simplify the Davis-Gray theory for van der Waals predissociation reactions. Specifically, they replaced the exact separatrix with an approximate phase space dividing surface by dropping a number of small terms in the system Hamiltonian, and they replaced the exact mapping that defines the flux across the tme separatrix with an analytic treatment of the flux across the approximate separatrix. This simplification is schematically presented in Fig. 18. 

To conclude this section we discuss the baker s map (Farmer et al. (1983)) as an example for an area preserving mapping in two dimensions. Area preservation is of utmost importance for Hamiltonian systems, since Liouville s theorem (Landau and Lifechitz (1970), Goldstein (1976)) guarantees the preservation of phase-space volume in the course of the time evolution of a Hamiltonian system. The baker s map is a transformation of the unit square onto itself. It is constructed in the following four steps illustrated in Fig. 2.5. 

In order to develop the criterion more quantitatively, consider the sequence of phase-space portraits shown in Figs. 5.4(a) - (d). This sequence suggests that, as the control parameter K increases, the diameter of the resonance islands at Z = 0 mod 27t grows in action. In order to predict the touching point of the resonances, we need the widths of the resonances as a function of K. The width of the resonances is derived on the basis of the Hamiltonian (5.2.1). Since the dynamics induced by H is equivalent to the chaotic mapping (5.1.6), the Hamiltonian H itself cannot be treated analytically and has to be simplified. One way is to consider only the average effect of the periodic 6 kicks in (5.2.1). The average perturbation... 

In this chapter we considered only a small Hamiltonian system whose Poincare map is the standard map defined on the unit square. It is interesting to consider Hamiltonian systems in a large phase space in which diffusion appears. Specifically, we are interested how the accelerator mode, which causes the anomalous diffusion in the standard map, affects the breaking of the adiabatic invariant. We will continue this study in a forthcoming article [21]. 

In two-dimensional Hamiltonian systems, the trajectories can be visualized by means of the Poincare surface of section plot. It is also possible to study two-dimensional Hamiltonian systems using the two-dimensional symplectic mapping. A typical phase space portrait of generic nonhyperbolic phase space is... 

This distinction between a < and a = exemplifies a broader theme in nonlinear dynamics. In general, if a map or flow contracts volumes in phase space, it is called dissipative. Dissipative systems commonly arise as models of physical situations involving friction, viscosity, or some other process that dissipates energy. In contrast, area-preserving maps are associated with conservative systems, particularly with the Hamiltonian systems of classical mechanics. 

The Poincare map is a method to transform the continuous flow in n-dimensional phase space to an equivalent discrete flow (map) in a phase space of (n — l)-dimensions (or (n — 2)-dimensions for Hamiltonian flows). 

The intersections of the continuous Hamiltonian flow in the 2n-dimensional phase space, defined by Equation (71), with the surface of section defined by Equation (72), transforms the continuous flow to an equivalent discrete flow (map), on a (2n — 2)-dimensional surface of section (see Figure 12). 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

Asymptotically, hyperspherical coordinates become inadequate since the energetically allowed space contains fewer and fewer grid points. It is therefore necessary to map the wave function onto other coordinates (e.g., Jacobi coordinates). However, in the semiclassical treatment of the problem this is not possible since the wave function is known only in a restricted phase space, i.e., in either (6, ) or (p, 0, < >) space. It is therefore necessary either to carry out the projection in these coordinates by using variable grid methodology or to introduce a mixed Jacobi-hyperspherical coordinate treatment. This latter procedure is possible since we can express the Hamiltonian as... 

The most straightforward rewriting of the Verlet method is to put the equations and the discretization in Hamiltonian form, i.e. introducing momenta/> = Mv, and thus pn = Mv , which results in the flow map approximation (taking us from any point in phase space (q,p) to a new point (Q, P) ... 

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

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